Full-wavefield seismic inversion (FWI) estimates a subsurface model by iteratively minimizing the difference between observed and simulated data. FIG. 7A illustrates the basic idea of full wavefield inversion of seismic data. The process is iterative, with most of the time being spent during the “compute gradient” step, which includes the very time-consuming computer simulation of predicted seismic data. The computer time and resource requirements for FWI are enormous. When FWI is applied using explicit time-domain simulations and iterative methods, the computational cost is proportional to the number of sources: this is the conventional sequential FWI approach, and it is illustrated in FIG. 7B for N source gathers of data. In a significant breakthrough in seismic data inversion, Krebs et al. (Geophysics 74, p. wcc177, (2009), and U.S. Pat. No. 8,121,823 to Krebs, et al.) described a simultaneous-source approach whereby many (or all) of the sources are encoded and then combined into a composite encoded source to greatly reduce the computational expense. Both Krebs et al. references are incorporated herein by reference in all jurisdictions that allow it. This simultaneous-source approach is illustrated in FIG. 7C. It relies on encoding multiple source gathers of data, generating what may be called a super shot made up of a weighted sum of individual shots, where the encoding functions are the weights, and then inverting the composite or “super” shot in a single inversion/simulation. If n shots are encoded and inverted simultaneously, this results in a speed-up by a factor of n. In one embodiment of the Krebs invention, the encoding functions are chosen randomly, for example the weights are either +1 or −1 chosen with equal probability. It is also known that other probabilistically chosen weights produce similar results. However, one super shot may not contain enough information, so the sum of individually simulated super shots—each with a different set of weights—may be used, as taught in the Krebs et al. (2009) reference. Each such simulation is called a realization and it is known that as the number of realizations increases, their sum will approximate the sum of simulations with the original shots, i.e. the sequential FWI approach. But this approximation will only be exact with an infinite number of realizations.
Godwin and Sava review a number of ways to produce encoding weights, including orthogonal weight vectors, in “A comparison of shot-encoding schemes for wave-equation migration,” Geophysical Prospecting, 1-18 (2013). However, they do not disclose the methods of selecting orthogonal weights that are disclosed in the following invention description. Moreover, their encoding is used for migration, which is not iterative and which does not improve the model.
There is a need for a method of choosing the weights deterministically so that realizations approximate the sequential FWI behavior as quickly as possible. The present invention satisfies this need.